The plant is there to make Leon more human
Such as the milk, the plant is a key element in that amazing movie. The plant is there to complete Leon'sJean Reno's character personality, providing consistence and humans features for him.
Indeed, Leon is a controversial character and his relationship with that plant is an aspect of significant relevance in order to helpincrease the identification between the viewer and Leonthe character: although he is a blood-cold hitman, Leon is also a human with sentimental needs, just as everyone. Without human features like esteem, fears and generosity it would be hard to gain the sympathy for that character.
The plant was a master choice. As noticed in other answers, the plant doesn't move, no words, no barking but still a living being and always ready to be loved. Therefore, a perfect match for the script.
FinallyIn addition, the plant is there also to interchange the continuity of scenes between Matilda and Leon almost as an another, a third, character.
Finally, the plant has no background story. At least, I never found one myself. Great scripts always leave those empty spaces so viewers can fill it themselves making personal connections with the plot.
PS.: This answer represents basically my personal opinion only.